The equations of the lines through the point (3, 2) which makes an angle of 45° with the line x - 2y = 3 are

Question

The equations of the lines through the point (3, 2) which makes an angle of $45 °$ with the line $x - 2 y = 3$ are

Smart Achievers · Accepted Answer

(1)

$Let the slope of the required line be m .$

$∴$ $\frac{y - 2}{x - 3} = m$ ...(i)

$Given line is$ $x - 2 y = 3 \Rightarrow y = \frac{1}{2} x - \frac{3}{2}$ ...(ii)

$Clearly, slope of the line is$ $\frac{1}{2}$

$It is given that the angle between (i) and (ii) is$ $45 °$

$∴$ $| \frac{m - \frac{1}{2}}{1 + \frac{1}{2} m} |$ $= \tan 45 °$ $\Rightarrow \frac{2 m - 1}{2 + m} = 1 or \frac{2 m - 1}{2 + m} = - 1$

$\Rightarrow$ $m = 3 or m = - \frac{1}{3}$

$∴$ $Required equation is \frac{y - 2}{x - 3} = 3 or \frac{y - 2}{x - 3} = - \frac{1}{3}$

$i.e.,$ $3 x - y - 7 = 0 or x + 3 y - 9 = 0 .$

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